Using the Calderón-Zygmund method of rotations and the uniform boundedness of the bilinear Hilbert transforms the authors show that a bilinear singular integral operator with rough kernel is bounded from \(L^p(\mathbb{R}) \times L^q(\mathbb{R})\) to \(L^r(\mathbb{R})\) for many indices satisfying the condition \(1/p + 1/q = 1/r\). Moreover, they provide an example of a function \(\Omega\) in \(L^q(S^1)\) with mean value zero such that the singular integral given by the convolution with p.v.\(\Omega(x/|x|)|x|^{-2}\) is not bounded from \(L^{p_1}(\mathbb{R}) \times L^{p_2}(\mathbb{R})\) to \(L^{p}(\mathbb{R})\) for \(1/22\).